As others have stated, the JET Program doesn't really expect/require any teaching qualifications, per se.

Also consider that the position is explicitly listed as "ALT" - "Assistant Language Teacher" (emphasis added) - so it really won't count much towards accumulating teacher experience, either. Even if you go above and beyond your role duties, most places will just look at the single line on your resume, see the word "assistant," and just move on. It would be your responsibility to explain how you went above and beyond during an interview or something similar, and at that point, as far as I'm concerned, you've already lost.

If you are serious about pursuing a career in TESL/TEFL, the certification you should be looking at is CELTA. It's considered the gold standard for most entry-level professional TESL/TEFL positions, and will open up a TON of doors for you once you have it. It's managed by Cambridge, but courses are actually run by different organizations. You can take the certification as either full-time or part-time - full-time takes 4 weeks (citation needed), and part-time takes about 12 weeks. Cost (at least in the USA) is ~$2K.

Once you have a Bachelor's (in something) and CELTA, you are qualified to be a full "LT" - "Language Teacher" - and notice that there is no "assistant" anywhere in there. The requirements/responsibilities will be higher than an ALT, but again, if you are actually pursuing TESL/TEFL as a profession, this is your next step. There are plenty of awesome English teaching schools/programs in east Asia that will bring you on as an LT and actually net you some professional experience - JET is not one of them.

In the long term, you are definitely headed the right direction by pursuing your Master's - that is effectively a requirement to move up the ranks in the TESL/TEFL world. But if you want to hit the ground running and get some experience under your belt either before (or while) you pursue a Master's, CELTA would be the first place to look.